# Ponder This

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## September 2018 - Solution

September 2018 Challenge

One cannot guarantee a mono-gender triplet with 4 people or less.
Five people (using the pigeon hole principle) guarantee a mono-gender triplet, but this requires 10 tries.
Adding a sixth person does not improve the solution, but a seventh one does.

We first bring Eden Saig's solution:
We start by observing that triplets of people that are being checked correspond to a 3-uniform hyper-graph: One can show that the problem of finding the shortest list of three letters words is equivalent to finding the smallest 3-uniform hyper-graph which is not 2-colorable (if the corresponding hyper-graph is 2 colorable, there is a gender assignment in which there is no triplet that is mono-gender).

A paper by Noga Alon from 1985 states that the minimal number of edges in a 3-uniform hyper-graph which is not 2-colorable is 7, and the graph which achieves lower this bound is the Fano plane.

To get the final list of words, we go over all combinations of 7 letters, assign them to the vertices of the Fano plane, and check whether all hyper-edges correspond to dictionary words under some permutation.

We received solutions in many languages:
Afrikaans (Tyler Mullen): UIE AUB DUS DIA BIS SAE BED
Danish (Pål Hermunn Johansen): ETA AIS NAP SEN PIE PST TIN
Dutch (David Greer): OEI PER SIP ROS POT RIT ETS
Finnish (Kang Jin Cho): KPL KEA PAI LEI KOI OPE LAO
French (Alex Fleischer): EAU NES NUL SAL TAN TEL TUS
German (Robert Lang): AUE ALT MAG GUT ULM MET GEL
Korean (Kang Jin Cho): ㅗㅏㅣ ㄱㅗㅇ ㅇㅏㄹ ㄱㅣㄹ ㅂㅗㄹ ㅂㅏㄱ ㅇㅣㅂ
Hebrew (Motty Porat): תיש עיט תטה שעה תער שטר הרי
Italian (Alper Halbutogullari): AHO AIE BEH BIO CAB CHI ECO
Japaneese (Vladimir Volevich) からい こわい うすい こうか わかす こらす わらう
Latin (Allen Zhu): TUA DAS SEU DIU SIT EIA DET
Norwegian (Pål Hermunn Johansen): BAN TAR ØYA BYR BØT ØRN NYT
Polish (Oscar Volpatti) PST OSA IKS POI KAP KTO TAI Portuguese (David Greer): EIA MAL LEU MUI VIL VEM UVA
Romanian (Radu-Alexandru Todor): FOC FIU FAR OUA ROI CAI URC Russian (Vladimir Volevich) гик снг рог ион рис рнк сок
Spanish (Allen Zhu): LAS CAE COL DEL CDS ODA ESO

Andreas Stiller even managed to find solution with IBM:
German: IBM dem reb dir dbs sie mrs
French: IBM gia gbl mal bas ils gsm

Paul Shaw sent us 91 solutions:

```doe duh dry hey rue rho you
hoe new yet nth why yon tow
opt sox you pus pyx sty tux
ate haw ani hie new nth wit
bud bye orb doe dry rue you
hoe tie men him nth ion tom
hug gym gos ohm shy sum you
cap aye air ice cry per yip
pad aye air die dry per yip
ado aye ran den dry ore yon
toe yep rue pro you put try
toe yep sue sop you put sty
got gyp rug pro you put try
aye gas mat get ems gym sty
ate hay ran her yen nth try
aye rag apt get per gyp try
aye air apt tie per yip try
aye jar oat jet ore joy try
aye ran oat net ore yon try
hay ran oat nth rho yon try
aye gas lam gel ems gym sly
bye hub sob hoe sue shy you
dye duh dos hoe sue shy you
aye haw art her wet thy wry
aye oaf ran fen ore fry yon
aye gas auk keg sue guy sky
aye ash mat hem set thy mys
aye ash maw hem sew why mys
aye ash oat hoe set thy soy
aye ash apt hep set thy spy
aye ash paw hep sew why spy
aye hat raw her wet why try
aye hat saw hes wet why sty
gym gos pug mop sum you spy
hey toe rue rho hut you try
yen toe rue nor nut you try
hey toe sue hos hut you sty
yen toe sue son nut you sty
den hid dot hoe tie nth ion
bah aye oat bet boy hoe thy
had aye raw dew dry her why
hie new toe nth how ion wit
get hue eon hog gnu nth out
bar ado aye bed boy dry ore
bet hob bun hue eon nth out
aye gal amp peg elm gym ply
aye hag mat get hem gym thy
aye hag apt get hep gyp thy
bye hob but hue toe thy you
hue eon pet nth hop pun out
hue eon set nth hos sun out
gos gyp gut opt you pus sty
hoe yen wet nth why now toy
aye gas amp peg ems gym spy
aye tag arm erg met gym try
aye tag par erg pet gyp try
aye wag arm erg mew gym wry
aye wag par erg pew gyp wry
bye bug mob ego emu gym you
dye dug mod ego emu gym you
cue con cry yen ore run you
due don dry yen ore run you
doe dun dry yen rue nor you
ego emu yep gym pug mop you
ego emu yet gym gut tom you
cog cup cry gyp rug pro you
hie pen toe nth hop ion pit
hoe sue yet shy hut sot you
due dot dry ore yet you rut
cue cot cry ore yet you rut
hue toe yep hop thy you put
hug got gyp hop thy you put
hue toe rye rho thy you rut
duh dot dry rho thy you rut
hug gos gyp hop shy you pus
gum gos gyp mop mys you pus
lug gos gyp lop sly you pus
aye ham sat hes met thy mys
aye ham saw hes mew why mys
aye air vat tie rev ivy try
bye orb but toe rue you try
bye sob but toe sue you sty
gym gos gut tom sum you sty
aye asp tax pet sex pyx sty
bar aye oat bet boy ore try
cob bye bur cue cry ore you
hog gym gut hum thy tom you
foe yen rue fun fry nor you
hoe yen set nth shy son toy
ego hey emu hug gym ohm you
hoe tie pen hip nth ion opt
```

The best "sentence" we could find is: "She saw pew? Aye! Why? Spy hap."